Filters Flashcards
Magnitude of the transfer function is
Directly proportional to the product of the zero vector lengths
Inversely proportional to the product of the pole vector lengths
Gain enhancement by poles |H(jw)| = K/d
Gain suppression by zeros |H(jw)| = Kd
What’s the ideal pass band for a low pass filter
The entire interval from [0,wc] where wc is the breakpoint of the filter
To enhance the gain at every point in [0,wc]
There must be individual poles corresponding to each of the frequencies in [0,wc]
Semicircular wall of continuous poles
Stephen Butterworth had some ideas on the best shape
A semi-circular arc gives a flat passband
The filter is ______ the closer to a continuum you get
More ideal
The number of points you use is called
The “order” of the filter
5th order BW filter
5 equally spaced points on a semicircle
2nd Order BW filter
2 equally spaced points on a semicircle
What kind of filters more closely approximate the ideal filter
Higher order
Pafnuty Chebyshev
Used semi-ellipses
Resulted in a sharper transition into the stopband
Resulted in a non-flat passband
Elliptic filters
Use poles and zeros to get even more extra sharp transitions, but with ripple in both the stopband and passband
Chebyshev type II
Poles are the inverse of the poles of the type I filter
Normalized filters
Filters with corner frequency set to 1 radian per second
Also called prototype filters
s—> s/wc
How did butterworth figure out the circle thing?
Was renowned for solving seemingly impossible mathematical problems
Started with a condition called ‘maximal flatness’ which demands that the first 2n-1 derivatives of the amplitude response are zero at zero
Only had one breakpoint at wc
To convert HPF to LPF
Make the substitution s—>wc/s to mirror transfer function about wc
The same as swapping capacitors for inductors and using the reciprocal of the table value
